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# Exponential sum working group

## Goal

Understand Bourgain-Chang's proof of bounds on exponential sums over
small multiplicative subgroups.

## Literature

As a starting point, we will use
Bourgain, J.(1-IASP); Chang, M.-C.(1-CAR)
Exponential sum estimates over subgroups and almost subgroups of $\Bbb Z\sb Q\sp *$, where $Q$ is composite with few prime factors.
Geom. Funct. Anal. 16 (2006), no. 2, 327--366.

(Or click here.)

## First meeting

Speaker: Pär Kurlberg

Time/place: friday 070223, 14.15-15.15, room 3733.

Topics: Some classical bounds on complete exponential sums, such as
Gauss and Kloosterman sums, the Weil bounds (aka the Riemann
hypothesis for curves). Bounds on incomplete sums (Polya-Vinogradov
and Burgess), and some applications (e.g. the size of the smallest
quadratic non-residue.)

## Second meeting

Speaker: Pär Kurlberg

Time/place: tuesday 070227, 10.15-12, room 3721.

Topic: We will start with an overview of the proof.

## Third meeting

Speaker: Pär Kurlberg

Time/place: tuesday 070306, 10.15-12, room 3721.

Topic: We will wrap up the sketch of the proof by finishing the
"statistical" multiplicative invariance. We will then go onto the
proof of proposition 2.1.

## Fourth meeting

Time/place: tuesday 070313, 10.15-12, room 3721.

Speaker: Pär Kurlberg

Topic: We will finish the proof of proposition 2.1 - how to use BGS'
(Balog-Gowers-Szemeredi) to go from "statistical" multiplicative
invariance to true multiplicative invariance.

## Fifth meeting

Time/place: wednesday 070314, 13.00-15.00, room 3733.

Speaker: Alexander Engström

Topic: On addititive combinatorics and the proof of
Balog-Gowers-Szemeredi.

## Sixth Meeting

Time/place: 10:15 - 12:00, Tuesday 20 March 2007, room 3721.

Speaker: Liangyi Zhao

Topic: I'll talk about bounds for character sums due to D. A. Burgess.
In the interest of time, I will not go through all the details of the
proof but present only the key points. Also, I will simply quote some
results of H. Hasse and A. Weil which are needed in the proof.

Reference:

D. A. Burgess, On Characters Sums and Primitive Roots, Proc. London
Math. Soc., No. 3, Vol. 12, 1962, pp. 179 - 192

Lecture notes